Dedicated to eiπ+1729e^{i\pi} + 1729

(n=1100002Sm+5Sn)1729\left(\sum_{n=1}^{10000} 2S_m + 5S_n\right)^{1729}

Let SnS_n and SmS_m be two natural numbers for arbitary natural numbers nn and mm : such that , Sn1=2Sm+1S_{n-1}=2S_{m+1}.Also 5mSm5^m | S_m , provided that (n,m)10000(n,m) \leq 10000.

Then evaluate above sumation modulo 10

Bonus : Generalize this.

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